Weyl type upper bounds on the number of resonances near the real axis for trapped systems
Stefanov, Plamen
Journées équations aux dérivées partielles, (2001), p. 1-16 / Harvested from Numdam
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We study semiclassical resonances in a box Ω(h) of height h N , N≫1. We show that the semiclassical wave front set of the resonant states (including the “generalized eigenfunctions”) is contained in the set 𝒯 of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator P # (h) with discrete spectrum the number of resonances in Ω(h) is bounded by the number of eigenvalues of P # (h) in an interval a bit larger than the projection of Ω(h) on the real line. As an application, we prove a Weyl type estimate of the number of resonances in Ω(h) in terms of the measure of 𝒯. We prove a similar estimate in case of classical scattering by a metric and obstacle.

@article{JEDP_2001____A13_0,
     author = {Stefanov, Plamen},
     title = {Weyl type upper bounds on the number of resonances near the real axis for trapped systems},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {2001},
     pages = {1-16},
     doi = {10.5802/jedp.597},
     mrnumber = {1843414},
     zbl = {01808689},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEDP_2001____A13_0}
}

Stefanov, Plamen. Weyl type upper bounds on the number of resonances near the real axis for trapped systems. Journées équations aux dérivées partielles,  (2001), pp. 1-16. doi : 10.5802/jedp.597. http://gdmltest.u-ga.fr/item/JEDP_2001____A13_0/

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